PUH Theorem 401 — PUH's Field Equation Under Stated Postulates: The Archive's Requirements Select Einstein's Equations or Their Unimodular Form, and the Lattice Must Still Supply the Carrier

Photonic Universe Hypothesis (PUH) — Theorem 401. The archive has lacked a field equation for gravity since T298's audit. T301 took the static one from general relativity as a constraint, T329 found that the lattice does not yet reproduce it, and T193's Theorem 4 stated, without derivation, that Einstein's equations are the E8 condensate's field equations. Since then T383, T384 and T385 have fixed what gravity's carrier must do: be a field with directions whose rest form is T299's metric, carry the rest frame of the matter that makes it, and radiate T193's two transverse patterns, fed by the stars' motion and by the lattice stretched between them, while its one-number part stays still. This paper asks which field equations those requirements select. Its results are standard results of the field theory of gravity (Fierz and Pauli 1939; Weinberg 1965; Deser 1970), recomputed in the archive's language and against its own requirements. Result 401.1: for an equation with two derivatives and T317's Lorentz symmetry, the requirements fix the field every conserved source makes and feels to be the linearised Einstein one — light at the Sun's limb bent 1.7512″, Newton's inverse-square pull, T383's tensor boost with T384's factor of 4, and two patterns at c with positive energy. Two equations give it with no further waves: the one unchanged by every smooth relabelling of the lattice's cells (linearised Einstein–Hilbert) and the one unchanged by volume-preserving relabellings with a shift of the one-number part (linearised unimodular gravity, coupled to the trace-free part of the source). For every choice of the four coefficients one number fixes the field of conserved sources, and only Einstein's value of it gives γ = 1; the other equations with that value add modes that matter never excites, some with negative energy or zero norm. Result 401.2: any restoring force on the tension is bounded. In the linearised theory a Lorentz-symmetric one bends light only 3/4 as much at any strength, though nonlinear completions can screen this within the Vainshtein radius, about 6 to 300 AU for the Sun; springs on the lattice's 240 bonds turn the pull into a push at 1.095 times their range; springs that resist only shear leave the static field unchanged. All make the waves massive, which GWTC-3 bounds to a range above 1.55 × 10¹⁶ m. Result 401.3: relabelling invariance forces the source to be conserved. Matter alone is not conserved in an orbiting pair; to Newtonian order the stress of T344's tension field restores conservation exactly, so it is a source with a share of exactly 1, and the double pulsar holds such a share within 1.3 × 10⁻⁴ of 1. The same invariance keeps the one-number part from radiating, so T385's OPEN (1) reduces to that invariance. Result 401.4: with no symmetry assumed, ten two-derivative terms are unchanged by relabelling, and some make Newton's constant depend on direction; the icosahedral group leaves two, both the same in every direction, so T396's icosahedral axes cannot make Newton's constant depend on direction at this order. The second marks the lattice's rest frame; GW170817 holds it below 6 × 10⁻¹⁵. Result 401.5: completed with its own stress, the equation is Einstein's (Deser 1970) or unimodular gravity — the equation T193's Theorem 4 named — selected by the archive's requirements, not yet derived from the lattice. What it means: IF the tension is the only long-range field coupled to matter and its equation is Lorentz-symmetric and relabelling-invariant, PUH's gravity at long wavelengths is Einstein's, and PUH's distinct predictions in gravity must come from the lattice scale and the Shell. Not claimed: that the results are new as physics; that the postulates follow from the lattice; anything about the Shell. OPEN: which degree of freedom of the E8 lattice is the tension — a tensor with no restoring force, blind to relabelling, whose bending costs c⁴/16πG = 2.41 × 10⁴² N — and which invariance the lattice gives it; Kleinert's crystal with only second-gradient elasticity and Sakharov's induced stiffness are published routes, not computed here, and the Weinberg–Witten theorem must be met. Kill-conditions are stated. Scripts and outputs are supplied.